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Number Systems. Computing Theory – F453. Data Representation. Data in a computer needs to be represented in a format the computer understands. This does not necessarily mean that this format is easy for us to understand. Not easy, but not impossible!

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number systems
Number Systems

Computing Theory – F453

data representation
Data Representation
  • Data in a computer needs to be represented in a format the computer understands.
  • This does not necessarily mean that this format is easy for us to understand.
  • Not easy, but not impossible!
  • A computer only understand the concept of ON and OFF.
  • Why?
  • How do we translate this into something WE understand?
  • We use a numeric representation (1s and 0s)
data representation1
Data Representation
  • If a computer can only understand ON and OFF, which is represented by 1 and 0, then which is which?
  • 1 = ON
  • 0 = OFF
  • This is known as the Binary system.
  • Because there are only 2 digits involved, it is known as Base 2.
  • But what does it MEAN??!
denary numbers
Denary Numbers
  • We use the Denary Number System.
  • This is in Base 10, because there are 10 single digits in our number system.
  • Why? We are surrounded by things that are divisible by ten.
  • Counting in tens is not a new phenomenon…
  • Even the Egyptians did it!
real numbers
Real Numbers
  • In the real world we have to work with decimal numbers, but there is no place in binary for a decimal point. In these cases, we need to ‘normalise’ the number.
  • In short, we place everything to the right of the decimal point:

Positive exponent represents the decimal point moving left

10111.0

0.10111 x 25

0.10111 x 2101

010111101

Exponent

Mantissa

real numbers1
Real Numbers
  • Example two:
  • The binary number 10.11011 with 8 bits for the mantissa and 6 for the exponent

.1011011

0. .1011011 x 22

0.10111 x 210

01011011000010

Positive exponent represents the decimal point moving left

Exponent

Mantissa

real numbers2
Real Numbers
  • Example two:
  • The denary number 37.5 with 8 bits for the mantissa and 6 for the exponent

37.5 =

100101.1

0.1001011 x 26

0.1001011 x 2110

01001011000110

Positive exponent represents the decimal point moving left

Exponent

Mantissa

real numbers your turn
Real Numbers – Your Turn
  • Example three:
  • The denary number 52.75 with 8 bits for the mantissa and 8 for the exponent

52.75 =

110100.11

0.11010011 x 26

0.11010011 x 2110

01101001100000110

Positive exponent represents the decimal point moving left

Exponent

Mantissa

real numbers your turn1
Real Numbers – Your Turn
  • Example four:
  • The denary number 22.8125 with 10 bits for the mantissa and 5 for the exponent

22.8125 =

010110.1101

0.101101101 x 25

0.101101101 x 2101

010110110101101

Positive exponent represents the decimal point moving left

Exponent

Mantissa

real numbers in reverse
Real Numbers – In reverse
  • Example:
  • The binary number 01011100000011 with 8 bits for the mantissa and 6 for the exponent

Mantissa

Exponent

01011100000011

0.1011100 x 211

0.1011100 x 23

0101.1100

= 5.75

real small numbers
Real Small Numbers
  • In the real world we also have to work with numbers which are less than 1, or decimals.
  • This is tackled in the same way, but we make use of two’s complement for the exponent:

Negative exponent represents the decimal point moving right

0.00010101

0.10101 x 2-3

0.10101 x 2-11

0.10101 x 2-11

01010111110101

Mantissa

Exponent

real numbers your turn2
Real Numbers – Your Turn
  • Example five:
  • The binary number 0.00110 with 8 bits for the mantissa and 8 for the exponent

0.00110

0.110 x 2-2

0.110 x 2-10

0.110 x 211111110

0000011011111110

Exponent

Mantissa

real numbers in reverse1
Real Numbers – In reverse
  • Example:
  • The binary number 01011110111010 with 8 bits for the mantissa and 6 for the exponent

Mantissa

Exponent

01011110111010

0.1011110 x 2-110

0.1011110 x 2-6

0.0000001011110

normalisation
Normalisation
  • In the examples, the point in the mantissa is always placed before the first zero (eg. 0.11010).
  • This not only allows for the maximum number to be held, but by ensuring that the first two digits are different, the mantissa is said to be normalised.
  • Therefore, for a positive number, the first digit is always a zero, and the exponent is held in two’s complement form.